An axiomatic description of a duality for modules (Q1369257)

A right module \(M\) over an associative ring with identity \(\Lambda\) is said to be simply reflexive provided there is a map \(X\to Y\) in the category \(\text{mod}(\Lambda)\) of all finitely presented right \(\Lambda\)-modules such that the cokernel of the induced map \(\Hom(Y,M)\to\Hom(X,M)\) is simple as \(\text{End}(M)^{op}\)-module. It is possible to assign a simply reflexive indecomposable pure-injective (SRIPI) left (resp. right) \(\Lambda\)-module \(DM\) to each SRIPI right (resp. left) \(\Lambda\)-module \(M\), and this assignment defines a bijection between the isomorphism classes of left and right SRIPI \(\Lambda\)-modules. Moreover, if \(I\) denotes an injective cogenerator for the center \(C\) of \(\Lambda\), then \(DM\) is isomorphic to a direct summand of \(\Hom_C(M,I)\). Examples of SRIPI modules are the indecomposable \(\Sigma\)-pure-injective modules. The construction of \(DM\) uses the well-known duality between the subcategories of finitely presented objects of the categories \({\mathcal D}(\Lambda)\) and \({\mathcal D}(\Lambda^{op})\) of additive functors from \(\text{mod}(\Lambda^{op})\) (resp. \(\text{mod}(\Lambda)\)) to the category of abelian groups [\textit{L. Gruson}, Lect. Notes Math. 488, 156-159 (1975; Zbl 0318.18012)]. The key is that if \(M\otimes_\Lambda-\) is regarded as an object of a certain quotient category of \({\mathcal D}(\Lambda)\), then it is the injective envelope of a simple object (quotient categories were studied by \textit{P. Gabriel} [Bull. Soc. Math. Fr. 90, 323-448 (1962; Zbl 0201.35602)]). The existence of such a simple object is guaranteed by the fact that \(M\) is simply reflexive. It is pointed out by the author: Our approach gives a new interpretation of a construction which is known as elementary duality amongst model theorists, and which is due to \textit{I. Herzog} [Trans. Am. Math. Soc. 340, No. 1, 37-69 (1993; Zbl 0815.16002)], see also [\textit{M. Prest, P. Rothmaler} and \textit{M. Ziegler}, J. Pure Appl. Algebra 93, No. 1, 33-56 (1994; Zbl 0801.03028)].